Similarity measures of bipolar neutrosophic sets and their application to multiple criteria decision

Neural Comput & Applic DOI 10.1007/s00521-016-2479-1

ORIGINAL ARTICLE

Similarity measures of bipolar neutrosophic sets and their application to multiple criteria decision making Vakkas Uluc¸ay1

•

Irfan Deli2 • Mehmet S¸ ahin1

Received: 5 March 2016 / Accepted: 6 July 2016 Ó The Natural Computing Applications Forum 2016

Abstract In this paper, we introduced some similarity measures for bipolar neutrosophic sets such as; Dice similarity measure, weighted Dice similarity measure, Hybrid vector similarity measure and weighted Hybrid vector similarity measure. Also we examine the propositions of the similarity measures. Furthermore, a multi-criteria decision-making method for bipolar neutrosophic set is developed based on these given similarity measures. Then, a practical example is shown to verify the feasibility of the new method. Finally, we compare the proposed method with the existing methods in order to demonstrate the practicality and effectiveness of the developed method in this paper.

Keywords Neutrosophic sets Bipolar neutrosophic set Dice similarity measure Hybrid vector similarity Decision making

1 Introduction As a generalization of classical sets, fuzzy set [51] and intuitionistic fuzzy set [3], neutrosophic set was presented

by Smarandache [32, 33] to capture the incomplete, indeterminate and inconsistent information. The neutrosophic set has three completely independent parts, which are truthmembership degree, indeterminacy-membership degree and falsity-membership degree; therefore, it is applied to & Vakkas Uluc¸ay vulucay27@gmail.com 1

Gaziantep University, 27310 Gaziantep, Turkey

2

Department of Mathematics, Kilis 7 Aralık University, 79000 Kilis, Turkey

many different areas, such as decision-making problems [1, 2, 4, 7–11, 19, 23, 31, 37–39, 42, 52]. In additionally, since the neutrosophic sets are hard to be applied in some real problems because of the truth-membership degree, indeterminacy-membership degree and falsity-membership degree lie in 0; 1þ ½, single-valued neutrosophic sets introduced by Wang et al. [40].

Recently, Lee [21, 22] proposed notation of bipolar fuzzy set and their operations based on fuzzy sets. A bipolar fuzzy set has two completely independent parts, which are positive membership degree T þ ! ½0; 1 and negative membership degree T ! ½ 1; 0 . Also the bipolar fuzzy models have been studied by many authors including theory and applications in [12, 17, 25, 34, 35, 50]. After the definition of Smarandache’s neutrosophic set, neutrosophic sets and neutrosophic logic have been applied in many real applications to handle uncertainty. The neutrosophic set uses one single value in 0; 1þ ½ to represent the truth-membership degree, indeterminacy-membership degree and falsitymembership degree of an element in the universe X. Then, Deli et al. [18] introduced the concept of bipolar neutrosophic sets, as an extension of neutrosophic sets. In the bipolar neutrosophic sets, the positive membership degree T þ ðxÞ; I þ ðxÞ; F þ ðxÞ denotes the truth membership, indeterminate membership and false membership of an element x 2 X corresponding to a bipolar neutrosophic set A and the negative membership degree T ðxÞ; I ðxÞ; F ðxÞ denotes the truth membership, indeterminate membership and false membership of an element x 2 X to some implicit counter-property corresponding to a bipolar neutrosophic set A. Similarity measure is an important tool in constructing multi-criteria decision-making methods in many areas such

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Neural Comput & Applic

as medical diagnosis, pattern recognition, clustering analysis, decision making and so on. Similarity measures under all sorts of fuzzy environments including single-valued neutrosophic environments have been studied by many researchers in [5, 6, 13, 14, 15, 20–24, 26–30, 43–49].

Also, Deli and S¸ ubas’s [16] and S¸ ahin et al. [36] presented some similarity measures on bipolar neutrosophic sets based on correlation coefficient similarity measure and Jaccard vector similarity measure of neutrosophic set and applied to a decision-making problem, respectively. This paper is constructed as follows. In Sect. 2, some basic definitions of neutrosophic sets and bipolar neutrosophic sets are introduced. In Sect. 3, we propose some similarity measures for bipolar neutrosophic sets such as; Dice similarity measure, weighted Dice similarity measure, Hybrid vector similarity measure and weighted Hybrid vector similarity measure and investigate their several properties. In Sect. 4, a multi-criteria decisionmaking method for bipolar neutrosophic set is developed based on these given similarity measures and a practical example is given. In Sect. 5, we compare the proposed method with the existing methods in order to demonstrate the practicality and effectiveness of the developed method in this paper. In Sect. 6, the conclusions are summarized.

membership function, respectively. They are respectively defined by

2 Preliminary

1.

In the section, we give some concepts related to neutrosophic sets and bipolar neutrosophic sets.

TA : E ! ½0; 1 ;

Definition 3 [18] A bipolar neutrosophic set(BNS) A in X is defined as an object of the form A ¼ fhx; T þ ðxÞ; I þ ðxÞ; F þ ðxÞ; T ðxÞ; I ðxÞ; F ðxÞi : x 2 Xg: where T þ ; I þ ; F þ : E ! ½0; 1 ; T ; I ; F : X ! ½ 1; 0 : The positive membership degree T þ ðxÞ; I þ ðxÞ; F þ ðxÞ denotes the truth membership, indeterminate membership and false membership of an element x 2 X corresponding to a bipolar neutrosophic set A and the negative membership degree T ðxÞ; I ðxÞ; F ðxÞ denotes the truth membership, indeterminate membership and false membership of an element x 2 X to some implicit counter-property corresponding to a bipolar neutrosophic set A. Definition 4 [18] Let A1 ¼ hx; T1þ ðxÞ; I1þ ðxÞ; F1þ ðxÞ; T1 ðxÞ; I1 ðxÞ; F1 ðxÞi and A2 ¼ hx; T2þ ðxÞ; I2þ ðxÞ; F2þ ðxÞ; T2 ðxÞ; I2 ðxÞ; F2 ðxÞi be two BNSs in a universe of discourse X. Then the following operations are defined as follows:

2.

3.

where TA ðxÞ, IA ðxÞ and FA ðxÞ are called truth-membership function, indeterminacy-membership function and falsitymembership function, respectively. They are, respectively, defined by

4.

TA : E ! 0; 1þ ½;

5.

IA : E ! 0; 1þ ½;

FA : E ! 0; 1þ ½

such that 0 TA ðxÞ þ IA ðxÞ þ FA ðxÞ 3þ .

Definition 2 [40] Let E be a universe. A single-valued neutrosophic set (SVN-set) A over E is defined by A ¼ fhx; ðTA ðxÞ; IA ðxÞ; FA ðxÞÞi : x 2 Eg: where TA ðxÞ, IA ðxÞ and FA ðxÞ are called truth-membership function, indeterminacy-membership function and falsity-

123

FA : E ! ½0; 1

such that 0 TA ðxÞ þ IA ðxÞ þ FA ðxÞ 3.

Definition 1 [32] Let E be a universe. A neutrosophic set A over E is defined by A ¼ fhx; ðTA ðxÞ; IA ðxÞ; FA ðxÞÞi : x 2 Eg:

IA : E ! ½0; 1 ;

A1 ¼ A2 if and only if T1þ ðxÞ ¼ T2þ ðxÞ; I1þ ðxÞ ¼ I2þ ðxÞ; F1þ ðxÞ ¼ F2þ ðxÞ and T1 ðxÞ ¼ T2 ðxÞ; I1 ðxÞ ¼ I2 ðxÞ; F1 ðxÞ ¼ F2 ðxÞ: A1 [ A2 ¼fhx; maxðT1þ ðxÞ; T2þ ðxÞÞ; I1þ ðxÞþI2þ ðxÞ ; minðF1þ ðxÞ; F2þ ðxÞÞ; minðT1 ðxÞ; T2 ðxÞÞ; 2

I1 ðxÞþI2 ðxÞ ; maxðF1 ðxÞ; F2 ðxÞÞig8x 2 X: 2 A1 \ A2 ¼fhx; minðT1þ ðxÞ; T2þ ðxÞÞ; I1þ ðxÞþI2þ ðxÞ ; maxðF1þ ðxÞ; F2þ ðxÞÞ; maxðT1 ðxÞ; 2 I ðxÞþI ðxÞ T2 ðxÞÞ; 1 2 2 ; minðF1 ðxÞ; F2 ðxÞÞig8x 2 X: Ac ¼ fhx; 1 TAþ ðxÞ; 1 IAþ ðxÞ; 1 FAþ ðxÞ; 1 TA ðxÞ; 1 IA ðxÞ; 1 FA ðxÞig A1 A2 if and only if T1þ ðxÞ T2þ ðxÞ; I1þ ðxÞ I2þ ðxÞ; F1þ ðxÞ F2þ ðxÞ and T1 ðxÞ T2 ðxÞ; I1 ðxÞ I2 ðxÞ; F1 ðxÞ F2 ðxÞ:

Definition 5 [45] Let A ¼ hTA ðxi Þ; IA ðxi Þ; FA ðxi Þi and B ¼ hTB ðxi Þ; IB ðxi Þ; FB ðxi Þi be two SVNSs in a universe of discourse X ¼ ðx1 ; x2 ; . . .; xn Þ: Then Dice similarity measure between SVNSs A and B in the vector space is defined as follows:

Neural Comput & Applic

DðA; BÞ ¼

n 1X

n i¼1

½ððTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi ÞÞ þ ððTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi ÞÞ

It satisfies the following properties: 0 DðA; BÞ 1; DðA; BÞ ¼ DðB; AÞ; DðA; BÞ ¼ 1 for A ¼ B i.e.TA ðxi Þ ¼ TB ðxi Þ; IA ðxi Þ ¼ IB ðxi Þ; FA ðxi Þ ¼ FB ðxi Þ ði ¼ 1; 2. . .; nÞ 8 xi ði ¼ 1; 2; . . .; nÞ 2 X:

1.

2.

3.

Definition 6 [29] Let A ¼ hTA ðxi Þ; IA ðxi Þ; FA ðxi Þi and B ¼ hTB ðxi Þ; IB ðxi Þ; FB ðxi Þi be two SVN-sets in a universe of discourse X ¼ ðx1 ; x2 ; . . .; xn Þ: Then

EðX; YÞ ¼ k

2ðTA ðxi ÞTB ðxi Þ þ IA ðxi ÞIB ðxi Þ þ FA ðxi ÞFB ðxi ÞÞ

3 Similarity measures of bipolar neutrosophic sets In this section, we introduce some similarity measures for bipolar neutrosophic sets including Dice similarity measure, weighted Dice similarity measure, Hybrid vector similarity measure and weighted Hybrid vector similarity measure by extending the studies in [29, 45].

Definition 7 Let A ¼ hTAþ ðxi Þ; IAþ ðxi Þ; FAþ ðxi Þ; TA ðxi Þ; IA ðxi Þ; FA ðxi Þi and B ¼ hTBþ ðxi Þ; IBþ ðxi Þ; FBþ ðxi Þ; TB ðxi Þ;

2ðTA ðxi ÞTB ðxi Þ þ IA ðxi ÞIB ðxi Þ þ FA ðxi ÞFB ðxi ÞÞ

½ððTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi ÞÞ þ ððTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi ÞÞ !

TA ðxi ÞTB ðxi Þ þ IA ðxi ÞIB ðxi Þ þ FA ðxi ÞFB ðxi Þ ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : þ ð1 kÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ ððTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi ÞÞ: ððTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi ÞÞ

It satisfies the following properties: 1.

2.

3.

0 DðA; BÞ 1; DðA; BÞ ¼ DðB; AÞ; DðA; BÞ ¼ 1 for A ¼ B i.e.

DðA; BÞ ¼

IB ðxi Þ; FB ðxi Þi be two BNSs in the set X ¼ fx1 ; x2 ; . . .; xn g: Then, Dice similarity measure between BNS A and B, denoted D(A, B), is defined as;

n 1X n i¼1 1 0 ½ðTAþ ðxi ÞTBþ ðxi Þ þ IAþ ðxi ÞIBþ ðxi Þ þ FAþ ðxi ÞFBþ ðxi ÞÞ ðTA ðxi ÞTB ðxi Þ þ IA ðxi ÞIB ðxi Þ þ FA ðxi ÞFB ðxi ÞÞ C B ½ððTAþ Þ2 ðxi Þ þ ðIAþ Þ2 ðxi Þ þ ðFAþ Þ2 ðxi ÞÞ þ ððTBþ Þ2 ðxi Þ þ ðIBþ Þ2 ðxi Þ þ ðFBþ Þ2 ðxi ÞÞ @ A

ððTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi ÞÞ ððTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi ÞÞ

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Neural Comput & Applic

Example 1 Suppose that A ¼ h0:5; 0:2; 0:6; 0:4; 0:3; 0:7i, B ¼ h0:7; 0:1; 0:3; 0:2; 0:2; 0:1i be two BNSs in the set. Then, 0 1 B DðA; BÞ ¼ @ 6

1.

2.

0 Dw ðA; BÞ 1; Dw ðA; BÞ ¼ Dw ðB; AÞ;

½ðð0:5Þð0:7Þ þ ð0:2Þð0:1Þ þ ð0:6Þð0:3ÞÞ ðð 0:4Þð 0:2Þ þ ð 0:3Þð 0:2Þ þ ð 0:7Þð 0:1ÞÞ ½ðð0:5Þ2 þ ð0:2Þ2 þ ð0:6Þ2 Þ þ ðð0:7Þ2 þ ð0:1Þ2 þ ð0:3Þ2 Þ ðð 0:4Þ2 þ ð 0:3Þ2 þ ð 0:7Þ2 Þ ðð 0:2Þ2 þ ð 0:2Þ2 þ ð 0:1Þ2 Þ

1 C A

¼ 0:1382

Definition 8 Let A ¼ hTAþ ðxi Þ; IAþ ðxi Þ; FAþ ðxi Þ; TA ðxi Þ; IA ðxi Þ; FA ðxi Þi and B ¼ hTBþ ðxi Þ; IBþ ðxi Þ; FBþ ðxi Þ; TB ðxi Þ; IB ðxi Þ; FB ðxi Þi be two BNSs in the set X ¼ fx1 ; x2 ; . . .; xn g and wi 2 ½0; 1 be the weight of each element xi for i ¼ Pn 1; 2; . . .; n such that i¼1 wi ¼ 1. Then, weighted Dice similarity measure between BNS A and B, denoted Dw ðA; BÞ, is defined as;

Dw ðA; BÞ ¼

n X

3.

Dw ðA; BÞ ¼ 1 for A ¼ B i.e.,TAþ ðxi Þ ¼ TBþ ðxi Þ; IAþ ðxi Þ ¼ IBþ ðxi Þ; FAþ ðxi Þ ¼ FBþ ðxi Þ; TA ðxi Þ ¼ TB ðxi Þ; IA ðxi Þ ¼ IB ðxi Þ; FA ðxi Þ ¼ FB ðxi Þ ði ¼ 1; 2. . .; nÞ 8 xi ði ¼ 1; 2; . . .; nÞ 2 X:

Proof 1.

It is clear from Definition 8.

wi

i¼1

0

B @

½ðTAþ ðxi ÞTBþ ðxi Þ þ IAþ ðxi ÞIBþ ðxi Þ þ FAþ ðxi ÞFBþ ðxi ÞÞ ðTA ðxi ÞTB ðxi Þ þ IA ðxi ÞIB ðxi Þ þ FA ðxi ÞFB ðxi ÞÞ ½ððTAþ Þ2 ðxi Þ þ ðIAþ Þ2 ðxi Þ þ ðFAþ Þ2 ðxi ÞÞ þ ððTBþ Þ2 ðxi Þ þ ðIBþ Þ2 ðxi Þ þ ðFBþ Þ2 ðxi ÞÞ ððTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi ÞÞ ððTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi ÞÞ

1 C A:

Example 2 Suppose that A ¼ h0:8; 0:2; 0:5; 0:2; 0:3; 0:4i, B ¼ h0:6; 0:4; 0:3; 0:1; 0:2; 0:3i be two BNSs in the set and w ¼ 0:3. Then, 0 B Dw ðA; BÞ ¼ 0:3 @

½ðð0:8Þð0:6Þ þ ð0:2Þð0:4Þ þ ð0:5Þð0:3ÞÞ ðð 0:2Þð 0:1Þ þ ð 0:3Þð 0:2Þ þ ð 0:4Þð 0:3ÞÞ ½ðð0:8Þ2 þ ð0:2Þ2 þ ð0:5Þ2 Þ þ ðð0:6Þ2 þ ð0:4Þ2 þ ð0:3Þ2 Þ ðð 0:2Þ2 þ ð 0:3Þ2 þ ð 0:4Þ2 Þ ðð 0:1Þ2 þ ð 0:2Þ2 þ ð 0:3Þ2 Þ

¼ 0:1378

Proposition 1 Let Dw ðA; BÞ be a weighted Dice similarity measure between BNSs A and B. Then, we have

123

1 C A

Neural Comput & Applic

2.

Dw ðA; BÞ ¼

n X

wi

i¼1

0

B @ ¼

n X

½ðTAþ ðxi ÞTBþ ðxi Þ þ IAþ ðxi ÞIBþ ðxi Þ þ FAþ ðxi ÞFBþ ðxi ÞÞ ðTA ðxi ÞTB ðxi Þ þ IA ðxi ÞIB ðxi Þ þ FA ðxi ÞFB ðxi ÞÞ ½ððTAþ Þ2 ðxi Þ þ ðIAþ Þ2 ðxi Þ þ ðFAþ Þ2 ðxi ÞÞ þ ððTBþ Þ2 ðxi Þ þ ðIBþ Þ2 ðxi Þ þ ðFBþ Þ2 ðxi ÞÞ ððTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi ÞÞ ððTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi ÞÞ

1 C A

wi

i¼1

0

B @

½ðTBþ ðxi ÞTAþ ðxi Þ þ IBþ ðxi ÞIAþ ðxi Þ þ FBþ ðxi ÞFAþ ðxi ÞÞ ðTB ðxi ÞTA ðxi Þ þ IB ðxi ÞIA ðxi Þ þ FB ðxi ÞFA ðxi ÞÞ ½ððTBþ Þ2 ðxi Þ þ ðIBþ Þ2 ðxi Þ þ ðFBþ Þ2 ðxi ÞÞ þ ððTAþ Þ2 ðxi Þ þ ðIAþ Þ2 ðxi Þ þ ðFAþ Þ2 ðxi ÞÞ ððTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi ÞÞ ððTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi ÞÞ

1 C A:

¼ Dw ðB; AÞ

3.

Since TAþ ðxi Þ ¼ TBþ ðxi Þ; IAþ ðxi Þ ¼ IBþ ðxi Þ; FAþ ðxi Þ ¼ FBþ ðxi Þ; TA ðxi Þ ¼ TB ðxi Þ; IA ðxi Þ ¼ IB ðxi Þ; FA ðxi Þ ¼ FB ðxi Þ ði ¼ 1; 2. . .; nÞ 8 xi ði ¼ 1; 2; . . .; nÞ 2 X, we have Dw ðA; BÞ ¼ 1. h

The proof is completed.

Definition 9 Let A ¼ hTAþ ðxi Þ; IAþ ðxi Þ; FAþ ðxi Þ; TA ðxi Þ; IA ðxi Þ; FA ðxi Þi and B ¼ hTBþ ðxi Þ; IBþ ðxi Þ; FBþ ðxi Þ; TB ðxi Þ; IB ðxi Þ; FB ðxi Þi be two BNSs in the set X ¼ fx1 ; x2 ; . . .; xn g.

Then, hybrid vector similarity measure between BNS A and B, denoted HybV(A, B), is defined as;

n 1X n i¼1 1 0 ½ðTAþ ðxi ÞTBþ ðxi Þ þ IAþ ðxi ÞIBþ ðxi Þ þ FAþ ðxi ÞFBþ ðxi ÞÞ ðTA ðxi ÞTB ðxi Þ þ IA ðxi ÞIB ðxi Þ þ FA ðxi ÞFB ðxi ÞÞ C B 2½ððTAþ Þ2 ðxi Þ þ ðIAþ Þ2 ðxi Þ þ ðFAþ Þ2 ðxi ÞÞ þ ððTBþ Þ2 ðxi Þ þ ðIBþ Þ2 ðxi Þ þ ðFBþ Þ2 ðxi ÞÞ @ A

HybVðA; BÞ ¼ k

þ ð1 kÞ 0 B B B @

ððTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi ÞÞ ððTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi ÞÞ n 1X n

i¼1 þ ½ðTA ðxi ÞTBþ ðxi Þ

þ IAþ ðxi ÞIBþ ðxi Þ þ FAþ ðxi ÞFBþ ðxi ÞÞ ðTA ðxi ÞTB ðxi Þ þ IA ðxi ÞIB ðxi Þ þ FA ðxi ÞFB ðxi ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2½ ðTAþ Þ2 ðxi Þ þ ðIAþ Þ2 ðxi Þ þ ðFAþ Þ2 ðxi Þ ðTBþ Þ2 ðxi Þ þ ðIBþ Þ2 ðxi Þ þ ðFBþ Þ2 ðxi Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi Þ ðTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi Þ

1 C C C A

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Neural Comput & Applic

Example 3 Suppose that A ¼ h0:6; 0:3; 0:4; 0:1; 0:3; 0:4i, B ¼ h0:5; 0:2; 0:3; 0:4; 0:2; 0:5i be two BNSs in the set. Then,

0 HybVðA; BÞ ¼ ð0:4Þ

1 B @ 6

HybVw ðA; BÞ ¼ 1 for A ¼ B i.e.,TAþ ðxi Þ ¼ TBþ ðxi Þ; IAþ ðxi Þ ¼ IBþ ðxi Þ; FAþ ðxi Þ ¼ FBþ ðxi Þ; TA ðxi Þ ¼ TB ðxi Þ; IA ðxi Þ ¼ IB ðxi Þ; FA ðxi Þ ¼ FB ðxi Þði ¼ 1; 2. . .; nÞ 8 xi ði ¼ 1; 2; . . .; nÞ 2 X:

½ðð0:6Þð0:5Þ þ ð0:3Þð0:2Þ þ ð0:4Þð0:3ÞÞ ðð 0:1Þð 0:4Þ þ ð 0:3Þð 0:2Þ þ ð 0:4Þð 0:5ÞÞ

0 þ ð1 ð0:4ÞÞ

3.

B 1 B B @ 6

1

C 2½ðð0:6Þ2 þ ð0:3Þ2 þ ð0:4Þ2 Þ þ ðð0:5Þ2 þ ð0:2Þ2 þ ð0:3Þ2 Þ A 2 2 2 2 2 2 ðð 0:1Þ þ ð 0:3Þ þ ð 0:4Þ Þ ðð 0:4Þ þ ð 0:2Þ þ ð 0:5Þ Þ ½ðð0:6Þð0:5Þ þ ð0:3Þð0:2Þ þ ð0:4Þð0:3ÞÞ ðð 0:1Þð 0:4Þ þ ð 0:3Þð 0:2Þ þ ð 0:4Þð 0:5ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2½ ðð0:6Þ2 þ ð0:3Þ2 þ ð0:4Þ2 Þ ðð0:5Þ2 þ ð0:2Þ2 þ ð0:3Þ2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðð 0:1Þ2 þ ð 0:3Þ2 þ ð 0:4Þ2 Þ ðð 0:4Þ2 þ ð 0:2Þ2 þ ð 0:5Þ2 Þ

1 C C C A

= 0.1421

Definition 10 Let A ¼ hTAþ ðxi Þ; IAþ ðxi Þ; FAþ ðxi Þ; TA ðxi Þ; IA ðxi Þ; FA ðxi Þi and B ¼ hTBþ ðxi Þ; IBþ ðxi Þ; FBþ ðxi Þ; TB ðxi Þ; IB ðxi Þ; FB ðxi Þi be two BNSs in the set X ¼ fx1 ; x2 ; . . .; xn g and wi 2 ½0; 1 be the weight of each element xi for i ¼ P 1; 2; . . .; n such that ni¼1 wi ¼ 1. Then, weighted hybrid vector similarity measure between BNS A and B, denoted HybVw ðA; BÞ, is defined as;

HybVw ðA; BÞ ¼ k

n X

4 BN-multi-criteria decision-making methods In this section, we introduce applications of weighted similarity measures in multi-criteria decision making problems under bipolar neutrosophic environment.

Definition 11 [36] Let U ¼ ðu1 ; u2 ; . . .; un Þ be a set of alternatives, A ¼ ða1 ; a2 ; . . .; am Þ be the set of attributes,

wi

i¼1

0

B @

½ðTAþ ðxi ÞTBþ ðxi Þ þ IAþ ðxi ÞIBþ ðxi Þ þ FAþ ðxi ÞFBþ ðxi ÞÞ ðTA ðxi ÞTB ðxi Þ þ IA ðxi ÞIB ðxi Þ þ FA ðxi ÞFB ðxi ÞÞ 2½ððTAþ Þ2 ðxi Þ þ ðIAþ Þ2 ðxi Þ þ ðFAþ Þ2 ðxi ÞÞ þ ððTBþ Þ2 ðxi Þ þ ðIBþ Þ2 ðxi Þ þ ðFBþ Þ2 ðxi ÞÞ ððTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi ÞÞ ððTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi ÞÞ

þ ð1 kÞ 0 B B B @

n X

½ðTAþ ðxi ÞTBþ ðxi Þ þ IAþ ðxi ÞIBþ ðxi Þ þ FAþ ðxi ÞFBþ ðxi ÞÞ ðTA ðxi ÞTB ðxi Þ þ IA ðxi ÞIB ðxi Þ þ FA ðxi ÞFB ðxi ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2½ ðTAþ Þ2 ðxi Þ þ ðIAþ Þ2 ðxi Þ þ ðFAþ Þ2 ðxi Þ ðTBþ Þ2 ðxi Þ þ ðIBþ Þ2 ðxi Þ þ ðFBþ Þ2 ðxi Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðTA Þ2 ðxi Þ þ ðIA Þ2 ðxi Þ þ ðFA Þ2 ðxi Þ ðTB Þ2 ðxi Þ þ ðIB Þ2 ðxi Þ þ ðFB Þ2 ðxi Þ

0 HybVw ðA; BÞ 1; HybVw ðA; BÞ ¼ HybVw ðB; AÞ;

123

C A

wi

i¼1

Proposition 2 Let HybVw ðA; BÞ be a weighted hybrid vector similarity measure between bipolar neutrosophic sets A and B. Then, we have 1.

2.

1

1 C C C A

w ¼ ðw1 ; w2 ; . . .; wn ÞT be the weight vector of the attributes P Cj ðj ¼ 1; 2; . . .; nÞ such that wj 0 and nj¼1 ¼ 1 and bij ¼ hTijþ ; Iijþ ; Fijþ ; Tij ; Iij ; Fij i be the decision matrix whose entries are the rating values of the alternatives. Then,

Neural Comput & Applic

[bij ]m×n

u1 u2 = .. .

⎛ ⎜ ⎜ ⎜ ⎝

um

a1 b11 b21 .. .

bm1

a2 b12 b22 .. .

bm2

··· ··· ··· .. .

an b1n b2n .. .

Algorithm

⎞ ⎟ ⎟ ⎟ ⎠

· · · bmn

is called an NB-multi-attribute decision-making matrix of the decision maker.

HybVw ðb j ; bi Þ ¼ k

n X

Step 1 Give the decision–making matrix ½bij m n ; for decision; Step 2 Compute the positive ideal (or negative ideal) bipolar neutrosophic solution for ½bij m n ;; Step 3 Calculate the weighted hybrid vector (or Dice) similarity measure between positive ideal (or negative ideal) bipolar neutrosophic solution b j and bi ¼ ½bij 1 n for all i ¼ 1; 2. . .; m and j ¼ 1; 2. . .; n as;

wj

j¼1

0

B B @

½ððTj Þþ ðTij Þþ þ ðIj Þþ ðIij Þþ þ ðFj Þþ ðFij Þþ Þ ððTj Þ ðTij Þ þ ðIj Þ ðIij Þ þ ðFj Þ ðFij Þ Þ 2½ðððTj Þþ Þ2 þ ððIj Þþ Þ2 þ ððFj Þþ Þ2 Þ þ ððTijþ Þ2 þ ðIijþ Þ2 þ ðFijþ Þ2 Þ ðððTj Þ Þ2 þ ððIj Þ Þ2 þ ððFj Þ Þ2 Þ ððTij Þ2 þ ðIij Þ2 þ ðFij Þ2 Þ

þ ð1 kÞ

n X

1 C C A

wj

j¼1

0 B B B @

½ððTj Þþ ðTij Þþ þ ðIj Þþ ðIij Þþ þ ðFj Þþ ðFij Þþ Þ ðTj Þ ðTij Þ þ ðIj Þ ðIij Þ þ ðFj Þ ðFij Þ Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2½ ððTj Þþ Þ2 þ ððIj Þþ Þ2 þ ððFj Þþ Þ2 ðTijþ Þ2 þ ðIijþ Þ2 þ ðFijþ Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððTj Þ Þ2 þ ððIj Þ Þ2 þ ððFj Þ Þ2 ðTij Þ2 þ ðIij Þ2 þ ðFij Þ2

Also; positive ideal bipolar neutrosophic solution u ¼ is the solution of decision matrix ½bij m n where every component of has the following form: ðb 1 ; b 2 ; . . .; b n Þ

b j ¼ hmaxi fTijþ g; mini fIijþ g; mini fFijþ g; mini fTij g; maxi fIij g; maxi fFij giðj ¼ 1; 2. . .; nÞ

ð1Þ

and negative ideal bipolar neutrosophic solution u ¼

ðb1 ; b2 ; . . .; bn Þ is the solution of decision matrix ½bij m n where every component of has the following form:

bj ¼ hmini fTijþ g; maxi fIijþ g; maxi fFijþ g; maxi fTij g; mini fIij g; mini fFij gi Now we give an algorithm as;

ð2Þ

1 C C C A

ð3Þ

Step 4. Determine the nonincreasing order of si ¼ HybVw ðb j ; bi Þ and select the best alternative.

Example 4 Let us consider the decision-making problem given in [41] for bipolar neutrosophic set. ‘‘Global environmental concern is a reality, and an increasing attention is focusing on the green production in various industries. A car company is desirable to select the most appropriate green supplier for one of the key elements in its manufacturing process.’’ [41]. After pre-evaluation, four suppliers(alternatives) are taken into consideration, which are denoted by u1 , u2 , u3 and u4 . Three criteria are considered including a1 is product quality; a2 is technology capability; a3 is pollution control. Assume the weight vector of the

123

Neural Comput & Applic

three criteria is W ¼ fw1 ; w2 ; w3 gT ¼ f0:2; 0:5; 0:3gT . In order to determine the decision information, an expert has gathered the criteria values for the four possible alternatives under bipolar neutrosophic environment. Then Algorithm Step 1 The decision–making matrix ½bij m n is given by a expert as; ⎛ u1 u2 ⎜ ⎜ u2 ⎝ u4

a1 0.4, 0.5, 0.3, −0.6, −0.4, −0.5 0.6, 0.4, 0.2, −0.4, −0.5, −0.7 0.7, 0.2, 0.4, −0.2, −0.6, −0.4 0.8, 0.6, 0.5, −0.5, −0.3, −0.6

a2 0.6, 0.1, 0.2, −0.4, −0.3, −0.2 0.6, 0.2, 0.3, −0.5, −0.2, −0.3 0.9, 0.3, 0.6, −0.2, −0.2, −0.5 0.6, 0.4, 0.3, −0.1, −0.3, −0.4

Step 2 The positive ideal bipolar neutrosophic solutions for are computed as; u ¼ ½h0:8; 0:2; 0:2; 0:6; 0:3; 0:4i; h0:9; 0:1; 0:2; 0:5; 0:2; 0:2i; h0:9; 0:1; 0:4; 0:5; 0:2; 0:1i : Step 3 The weighted hybrid vector similarity measure between positive ideal (or negative ideal) bipolar neutrosophic solution for alternative uj 2 U are computed and selected the best alternative.

Step 4 Ranking the alternatives (Table 1).

5 Comparative analysis and discussion In this subsection, a comparative study is presented to show the flexibility and feasibility of the introduced NB-multiattribute decision-making method. Therefore, different methods are used to solve the same NB-multi-attribute Table 1 Results for different values of k

decision-making problem with the bipolar neutrosophic information given by S¸ahin et al. [36], Deli et al. [18], Deli and S¸ ubas¸ [16]. S¸ ahin et al. [36] present a method by using Jaccard vector similarity and weighted Jaccard vector similarity measure and Deli and S¸ ubas¸ [16] present a method by using correlation measure based on multi-criteria decision making for bipolar neutrosophic sets. Also,

a3 0.8, 0.6, 0.5, −0.3, −0.2, −0.1 0.7, 0.4, 0.5, −0.1, −0.3, −0.4 0.6, 0.1, 0.5, −0.2, −0.4, −0.6 0.9, 0.6, 0.4, −0.5, −0.3, −0.6

⎞ ⎟ ⎟ ⎠

Deli et al. [18] contains two major phrases. The method firstly use score, certainty and accuracy functions to compare the bipolar neutrosophic sets. Secondly, he use bipolar neutrosophic weighted average operator and bipolar neutrosophic weighted geometric operator to aggregate the bipolar neutrosophic information. The ranking results obtained by different methods are summarized in Table 2.

In Table 2, there are some differences between the ranking results obtained by the methods. The optimal alternative is u3 obtained by the proposed methods except the result obtained by the method of Deli et al.’s method [18] and the proposed method HybVw with k ¼ 0:9. The reason may be score, certainty and accuracy functions and weighted average operator and bipolar neutrosophic weighted geometric operator in Deli et al.’s method [18] and parameter k in the proposed method HybVw . Generally, the proposed methods can effectively overcome the

Similarity measure

Values

Measure value

Ranking order

HybVw ðu ; ui Þ

k ¼ 0:25

HybVw ðu ; u1 Þ ¼ 0:24683

u3 u1 u4 u2

HybVw ðu ; u2 Þ ¼ 0:117780 HybVw ðu ; u3 Þ ¼ 0:27833 HybVw ðu ; u4 Þ ¼ 0:21136

HybVw ðu ; ui Þ

k ¼ 0:3

HybVw ðu ; u1 Þ ¼ 0:27063

u3 u1 u4 u2

HybVw ðu ; u2 Þ ¼ 0:19497 HybVw ðu ; u3 Þ ¼ 0:30222 HybVw ðu ; u4 Þ ¼ 0:22904 HybVw ðu ; ui Þ

k ¼ 0:6

HybVw ðu ; u1 Þ ¼ 0:41342

u3 u1 u4 u2

HybVw ðu ; u2 Þ ¼ 0:29803 HybVw ðu ; u3 Þ ¼ 0:44555 HybVw ðu ; u4 Þ ¼ 0:33510

HybVw ðu ; ui Þ

k ¼ 0:9

HybVw ðu ; u1 Þ ¼ 0:55620 HybVw ðu ; u2 Þ ¼ 0:40109 HybVw ðu ; u3 Þ ¼ 0:54313 HybVw ðu ; u4 Þ ¼ 0:44116

123

u1 u3 u4 u2

Neural Comput & Applic Table 2 The ranking results of different methods Methods

Ranking results

The proposed method HybVw with

k ¼ 0:25

u3 u1 u4 u2

The proposed method HybVw with

k ¼ 0:3

u3 u1 u4 u2

The proposed method HybVw with

k ¼ 0:6

u3 u1 u4 u2

The proposed method HybVw with

k ¼ 0:9

u1 u3 u4 u2

The proposed method Dw ðu ; ui Þ

u3 u1 u4 u2

Deli and S¸ ubas’s method [16]

u3 u1 u4 u2

Deli et al:0 s method [18]

u2 u3 u4 u1

S¸ ahin et al.0 s method [36]

u3 u4 u2 u1

decision-making problems which contain bipolar neutrosophic information. So, we think the proposed methods developed in this paper is more suitable to handle this application example.

6 Conclusion This paper developed a multi-criteria decision-making method for bipolar neutrosophic set is developed based on these given similarity measures. To get the comprehensive values, some similarity measures for bipolar neutrosophic sets such as; Dice similarity measure, weighted Dice similarity measure, Hybrid vector similarity measure and weighted Hybrid vector similarity measure are introduced.

In the future, it shall be significant to research some special kinds of bipolar neutrosophic measures.

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